Assume that the coefficients and solutions of the equation f (n)+p n−1(z)f (n−1)+. . .+p s+1(z)f (s+1)+ . . . + p 0(z)f = 0 have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients p j , j = s+1, . . . ,n−1, increase slower (in terms of the Nevanlinna characteristics) than p s (z). It is proved that this equation has at most s linearly independent solutions of finite order.
References
B. L. van der Waerden, Algebra [Russian translation], Nauka, Moscow (1976).
A. A. Gol’dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).
A. A. Mokhon’ko, “Malmquist theorem for the solutions of differential equations in a neighborhood of a logarithmic singular point,” Ukr. Mat. Zh., 56, No. 4, 476–483 (2004); English translation: Ukr. Math. J., 56, No. 4, 577–585 (2004).
Sh. I. Strelets, Asymptotic Properties of Analytic Solutions of Differential Equations [in Russian], Mintas, Vilnius (1972).
V. V. Golubev, Lectures on the Analytic Theory of Differential Equations [in Russian], Gostekhteorizdat, Moscow (1950).
M. Frei, “Uber die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten,” Comment. Math. Helv., 35, 201–222 (1961).
A. A. Mokhon’ko and A. Z. Mokhon’ko, “Deficiency values for the solutions of differential equations with branching point,” Ukr. Mat. Zh., 66, No. 7, 939–957 (2014); English translation: Ukr. Math. J., 66, No. 7, 1048–1069 (2014).
P. I. Lizorkin, A Course in Differential and Integral Equations with Additional Chapters of Analysis [in Russian], Nauka, Moscow (1981).
A. Z. Mokhon’ko, “A field of algebroidal functions and estimation of their Nevanlinna characteristics,” Sib. Mat. Zh., 22, No. 3, 213–218 (1981).
A. Z. Mokhon’ko, “An estimate of the modulus of the logarithmic derivative of a function which is meromorphic in an angular region, and its application,” Ukr. Mat. Zh., 41, No. 6, 839–843 (1989); English translation: Ukr. Math. J., No. 6, 722–725 (1989).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 139, pp. 139–144, January, 2014.
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Mokhon’ko, A.A., Mokhon’ko, A.Z. On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point. Ukr Math J 67, 159–165 (2015). https://doi.org/10.1007/s11253-015-1071-7
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DOI: https://doi.org/10.1007/s11253-015-1071-7